Abstract

A class of BAM neural networks with variable coefficients and neutral delays are investigated. By employing fixed-point theorem, the exponential dichotomy, and differential inequality techniques, we obtain some sufficient conditions to insure the existence and globally exponential stability of almost periodic solution. This is the first time to investigate the almost periodic solution of the BAM neutral neural network and the results of this paper are new, and they extend previously known results.

1. Introduction

Neural networks have been extensively investigated by experts of many areas such as pattern recognition, associative memory, and combinatorial optimization, recently, see [1–10]. Up to now, many results about stability of bidirectional associative memory (BAM) neural networks have been derived. For these BAM systems, periodic oscillatory behavior, almost periodic oscillatory properties, chaos, and bifurcation are their research contents; generally speaking, almost periodic oscillatory property is a common phenomenon in the real world, and in some aspects, it is more actual than other properties, see [11–21].

Time delays cannot be avoided in the hardware implementation of neural networks because of the finite switching speed of amplifiers and the finite signal propagation time in biological networks. The existence of time delay may lead to a system’s instability or oscillation, so delay cannot be neglected in modeling. It is known to all that many practical delay systems can be modelled as differential systems of neutral type, whose differential expression concludes not only the derivative term of the current state, but also concludes the derivative of the past state. It means that state’s changing at the past time may affect the current state. Practically, such phenomenon always appears in the study of automatic control, population dynamics, and so forth, and it is natural and important that systems will contain some information about the derivative of the past state to further describe and model the dynamics for such complex neural reactions [22]. Authors in [18–29] added neutral delay into the neural networks. In these papers, only [18–20] studied the almost periodic solution of the neutral neural networks. For example, in [19] the following network was studied:
Some sufficient conditions are obtained for the existence and globally exponential stability of almost periodic solution by employing fixed-point theorem and differential inequality techniques. References [21–26] studied the global asymptotic stability of equilibrium point, where [22] investigated the equilibrium point of the following BAM neutral neural network with constant coefficients:
By using the Lyapunov method and linear matrix inequality techniques, a new stability criterion was derived. References [27–29] studied the exponential stability of equilibrium point.

It is obviously that men always studied the stability of the equilibrium point of the neutral neural networks, and there is little result for the almost periodic solution of neutral neural networks, especially, for the BAM neutral type neural networks. Besides, in papers [11, 23, 27, 28], time delay must be differentiable, and its derivative is bounded, which we think is a strict condition.

Motivated by the above discussions, in this paper, we consider the almost periodic solution of a class of BAM neural networks with variable coefficients and neutral delays. By fixed-point theorem and differential inequality techniques, we obtain some sufficient conditions to insure the existence and globally exponential stability of almost periodic solution. To the best of the authors’ knowledge, this is the first time to investigate the almost periodic solution of the BAM neutral neural network, and we can remove delay’s derivable condition, so the results of this paper are new, and they extend previously known results.

2. Preliminaries

In this paper, we consider the following system:
where ; . , are the states of the th neuron of layer and the th neuron of layer, respectively; , and , are the delayed strengths of connectivity and the neutral delayed strengths of connectivity, respectively; , , , are activation functions; , stands for the external inputs; , , , and correspond to the delays, they are nonnegative; , represent the rate with which the th neuron of layer and the th neuron of layer will reset its potential to the resting state in isolation when disconnected from the networks.

Throughout this paper, we assume the following., , , , , , , , , , , and are continuous almost periodic functions. Moreover, we let
, , , and are Lipschitz continuous with the Lipschitz constants , , , , and .Consider

The initial conditions of system (2.1) are of the following form:
where ; ; , are continuous almost periodic functions.

Let where , are continuously differentiable almost periodic functions. For any , . We define , where , and is the derivative of at . Let , then is a Banach space.

The following definitions and lemmas will be used in this paper.

Definition 2.1 (see [11]). Let be continuous in . is said to be almost periodic on , if for any , the set is relatively dense, that is, for all , it is possible to find a real number , for any interval length , there exists a number in this interval such that , for all .

Definition 2.2 (see [11]). Let and be continuous matrix defined on . The following linear system:
is said to admit an exponential dichotomy on if there exist constants , , projection , and the fundamental solution of (2.5) satisfying

Definition 2.3. Let be a continuously differentiable almost periodic solution of (2.1) with initial value . If there exist constants , such that for every solution = = of (2.1) with any initial value = = , if
where , and are almost periodic functions. Then is said to be globally exponentially stable.

Lemma 2.4 (see [11]). If the linear system (2.5) admits an exponential dichotomy, then the almost periodic system
has a unique almost periodic solution

Lemma 2.5 (see [11]). Let be an almost periodic function on and
then the linear system admits exponential dichotomy on .

3. Existence and Uniqueness of Almost Periodic Solutions

In this section, we consider the existence and uniqueness of almost periodic solutions by fixed-point theorem.

Theorem 3.1. Under the assumptions –, the system (2.1) has a unique almost periodic solution in the region . If
holds, where

Proof. For any , we consider the the following system:
From () and Lemma 2.5, we know the following linear system:
admits an exponential dichotomy on . By Lemma 2.4, System (3.3) has an almost periodic solution which can be expressed as follows:
where
So, we can define a mapping , by letting
Set ; clearly, is a closed convex subset of , so we have
Therefore,
First, we prove that the mapping is a self-mapping from to . In fact, for any , let
From () and (), we have
This implies that , so is a self-mapping from to .Finally, we prove that is a contraction mapping. In fact, for any , . Let
We have
Notice that , it means that the mapping is a contraction mapping. By Banach fixed-point theorem, there exists a unique fixed-point such that , which implies system (2.1) has a unique almost periodic solution.

4. Global Exponential Stability of the Almost Periodic Solution

In this section, we consider the exponential stability of almost periodic solution, and we give two corollaries.

Theorem 4.1. Under the assumptions –, then system (2.1) has a unique almost periodic solution which is global exponentially stable.

Proof. It follows from Theorem 3.1 that system (2.1) has a unique almost periodic solution with the initial value = = . Set = = is an arbitrary solution of system (2.1) with initial value = = . Let , , , . Then = , where ; . Then system (2.1) is equivalent to the following system:
with the initial value
where
Let
where , , . From , we know , . Since and are continuous on and , as , , so there exist , such that and for , for . By choosing , we obtain , . So we can choose a positive constant , such that , . For the same reason, we define
There exists , , , such that , . Taking , since , , , and are strictly monotonous decrease functions, therefore, , , , , which implies
Multiplying the two equations of system (4.1) by and , respectively, and integrating on , we get
Taking
then , thus
where as in (4.6). We claim that
To prove (4.10), we first show for any , the following inequality holds:
If (4.11) is false, then there must be some and some , , such that
By (4.3)–(4.8), (4.12), and (4.13), we have
We also can get
From (4.14)–(4.15), we have
which contradicts the equality (4.12), so (4.11) holds. Letting , then (4.10) holds. The almost periodic solution of system (2.1) is globally exponentially stable.

Corollary 4.2. Let . Under assumptions , , and , if,
holds, then system
has a unique almost periodic solution in the region , which is global exponentially stable.

In fact, Zhang and Si [11, 16] and Chen et al. [17] studied system (4.18). This Corollary 4.2 is the Theorem 3.1 in [11], Theorem 1.1 in [16], and Theorem 1 in [17]. Especially, in [17], authors letTherefore, we extend and improve previously known results.

Remark 4.3. Let , , , , , , . Then system (2.1) is reduced to be system (1.1), hence we have the following.

Corollary 4.4. Under assumptions , , and , if
holds, then system (1.1) has a unique almost periodic solution in the region , which is global exponentially stable.

5. An Example

In this section, we give an example to illustrate the effectiveness of our results.

Let , , , , , , , , , , and , then we consider the following almost periodic system:
where , , , , , , , , , , , , , , , , , , , and . By simple calculation, we obtain , hence this system has a unique almost periodic solution, which is global exponentially stable by Theorem 4.1. Figure 1 depicts the time responses of state variables of , , , and with step and initial states for , and Figures 2, 3, and 4 depict the phase orbits of and , , and , and . It confirms that our results are effective for (5.1).

Figure 1: Transient response of state variable , , and .

Figure 2: Phase response of state variable and .

Figure 3: Phase response of state variable and .

Figure 4: Phase response of state variable and .

6. Conclusions

In this paper, a class of BAM neural networks with variable coefficients and neutral time-varying delays are investigated. By employing Banach fixed-point theorem, the exponential dichotomy and differential inequality techniques, some sufficient conditions are obtained to ensure the existence, uniqueness, and stability of the almost periodic solution. As is known to all, neural networks with neutral delays are studied rarely, and most authors solve these problems by linear matrix inequality techniques. In addition, BAM neural networks are much more complicated than the one-layer neural network. In a word, this paper is original, and novel. It also extends and improves other previously known results (see [11, 16, 17, 19]).

Acknowledgment

This work is supported by Distinguished Expert Science Foundation of Naval Aeronautical and Astronautical University.